Friday, May 9, 2014

GTO Brainteaser #5 -- Bayesian Werewolves

When a citizen of the ancient city of Bayes is accused of being a werewolf they are brought before the Tribunal to be considered for execution. The three Tribunal members can detect if someone is a werewolf or not through a simple spiritual ritual involving steamed badger milk. Since the ritual is only 90% accurate (yes badger milk is actually 90% effective for werewolf detection), each Tribunal member performs it separately, in secret and then decides to vote guilty or innocent. The accused is only executed if the Tribunal unanimously votes guilty. The wise Tribunal members are unbiased and go into each trial believing that there is a 50%/50% chance that the accused is a werewolf prior to conducting their ritual, however the members base their vote 100% on strategic self interest.

If the citizen is executed and reverts to wolf-form upon death, they were a werewolf and the Tribunal is given the accused’s possessions for their wisdom and public service.

If they do not turn into a wolf upon death, the Tribunal has executed an innocent citizen, and must each pay the citizen’s family a grievance fee. The family also gets a cake that says “Sorry guys, our bad, #sorrynotsorry”.

When the Tribunal sets a non-werewolf free, usually not much happens. The people of Bayes eat some discarded apology cake, get drunk, and think of how it might be fun to accuse other people of being werewolves.

If the Tribunal lets a werewolf go free, this is revealed when they turn into a wolf upon death (often at the hands of a cake-filled, drunken mob under a full moon). As a penance,the Tribunal pays the werewolf’s family the grievance fee and gets none of the werewolf’s possessions.

Imagine that you are one of the members of the Tribunal of Bayes presiding over the fate of one of the richest men in town. If you all vote to execute him, and are correct, you will each get 1000 gold coins (which buys a lot of cake in Bayes). If you unanimously execute him, and he was innocent, or if you let him go, and he is later found to be a werewolf, you must each pay a 200 gold coin fee. If you correctly set him free, you gain/lose nothing.

What is an equilibrium (GTO) strategy for voting based on the result of the ritual? What is the expected value in gold coins for each tribunal member when they all follow the equilibrium voting strategy, and what is the probability that they convict an innocent citizen or that they release a guilty citizen? How would these numbers change if the Tribunal used majority rule rather than requiring unanimity?EDIT: To clarify the efficacy of the badger milk ritual. It is 90% accurate in both directions. That is, if the accused is a werewolf there is a 10% chance the badger milk ritual will say that he is not, and if the accused is not a werewolf there is a 10% chance that the badger milk ritual will say that he is.